Integrand size = 22, antiderivative size = 99 \[ \int \frac {x \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {3 d \sqrt {c+d x^2}}{2 b^2}-\frac {\left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}-\frac {3 d \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 b^{5/2}} \]
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Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {455, 43, 52, 65, 214} \[ \int \frac {x \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=-\frac {3 d \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 b^{5/2}}-\frac {\left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac {3 d \sqrt {c+d x^2}}{2 b^2} \]
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Rule 43
Rule 52
Rule 65
Rule 214
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {(c+d x)^{3/2}}{(a+b x)^2} \, dx,x,x^2\right ) \\ & = -\frac {\left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac {(3 d) \text {Subst}\left (\int \frac {\sqrt {c+d x}}{a+b x} \, dx,x,x^2\right )}{4 b} \\ & = \frac {3 d \sqrt {c+d x^2}}{2 b^2}-\frac {\left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac {(3 d (b c-a d)) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right )}{4 b^2} \\ & = \frac {3 d \sqrt {c+d x^2}}{2 b^2}-\frac {\left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}+\frac {(3 (b c-a d)) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{2 b^2} \\ & = \frac {3 d \sqrt {c+d x^2}}{2 b^2}-\frac {\left (c+d x^2\right )^{3/2}}{2 b \left (a+b x^2\right )}-\frac {3 d \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 b^{5/2}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.97 \[ \int \frac {x \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {c+d x^2} \left (-b c+3 a d+2 b d x^2\right )}{2 b^2 \left (a+b x^2\right )}-\frac {3 d \sqrt {-b c+a d} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{2 b^{5/2}} \]
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Time = 3.06 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.10
method | result | size |
pseudoelliptic | \(\frac {-\frac {3 d \left (b \,x^{2}+a \right ) \left (a d -b c \right ) \arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2}+\frac {3 \sqrt {\left (a d -b c \right ) b}\, \sqrt {d \,x^{2}+c}\, \left (\frac {\left (2 d \,x^{2}-c \right ) b}{3}+a d \right )}{2}}{b^{2} \left (b \,x^{2}+a \right ) \sqrt {\left (a d -b c \right ) b}}\) | \(109\) |
risch | \(\frac {d \sqrt {d \,x^{2}+c}}{b^{2}}-\frac {-\frac {\left (a d -b c \right ) d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -b c}{b}}}-\frac {\left (a d -b c \right ) d \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -b c}{b}}}-\frac {\sqrt {-a b}\, \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {b \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x +\frac {\sqrt {-a b}}{b}\right )}+\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 a \,b^{2}}+\frac {\sqrt {-a b}\, \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {b \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{\left (a d -b c \right ) \left (x -\frac {\sqrt {-a b}}{b}\right )}-\frac {d \sqrt {-a b}\, \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{\left (a d -b c \right ) \sqrt {-\frac {a d -b c}{b}}}\right )}{4 a \,b^{2}}}{b^{2}}\) | \(902\) |
default | \(\text {Expression too large to display}\) | \(2152\) |
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Time = 0.28 (sec) , antiderivative size = 333, normalized size of antiderivative = 3.36 \[ \int \frac {x \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\left [\frac {3 \, {\left (b d x^{2} + a d\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (2 \, b d x^{2} - b c + 3 \, a d\right )} \sqrt {d x^{2} + c}}{8 \, {\left (b^{3} x^{2} + a b^{2}\right )}}, -\frac {3 \, {\left (b d x^{2} + a d\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) - 2 \, {\left (2 \, b d x^{2} - b c + 3 \, a d\right )} \sqrt {d x^{2} + c}}{4 \, {\left (b^{3} x^{2} + a b^{2}\right )}}\right ] \]
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\[ \int \frac {x \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {x \left (c + d x^{2}\right )^{\frac {3}{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {x \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.23 \[ \int \frac {x \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {d x^{2} + c} d}{b^{2}} + \frac {3 \, {\left (b c d - a d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} b^{2}} - \frac {\sqrt {d x^{2} + c} b c d - \sqrt {d x^{2} + c} a d^{2}}{2 \, {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )} b^{2}} \]
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Time = 5.51 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.18 \[ \int \frac {x \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^2} \, dx=\frac {\sqrt {d\,x^2+c}\,\left (\frac {a\,d^2}{2}-\frac {b\,c\,d}{2}\right )}{b^3\,\left (d\,x^2+c\right )-b^3\,c+a\,b^2\,d}+\frac {d\,\sqrt {d\,x^2+c}}{b^2}-\frac {3\,d\,\mathrm {atan}\left (\frac {\sqrt {b}\,d\,\sqrt {d\,x^2+c}\,\sqrt {a\,d-b\,c}}{a\,d^2-b\,c\,d}\right )\,\sqrt {a\,d-b\,c}}{2\,b^{5/2}} \]
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